Question: Let E be a Banach space and let $(x_n)$ be a sequence in $E$ such that $x_n\rightharpoonup x$ in the weak topology $\sigma(E, E')$. Prove that there exists a sequence $(z_n) \in E$ such that
- $z_n \in \text{conv}\left(\displaystyle\bigcup_{i=1}^{n}\{x_i\}\right)\; \forall n$, and
- $z_n\longrightarrow x$ strongly.
I'm strugglin to prove this question, I guess I'm misunderstanding some new concepts in the weak topology. But, I've reviwed in the Brezis'Book, and I still in doubt. Would someone give a hint here?
